When n gets larger and larger, Sn gets closer and
closer to the number 2. When a sequence Sn
gets closer and closer and closer to a given number S,
we say that S is the limit of the Sn's
and we write

lim( Sn ) = S

To take a physical analogy, consider a student who is one yard
from the wall of the classroom. He takes a large step to cut
the distance to the wall in half. Then he takes another
step to cut the distance in half again. He repeates this again
and again, getting closer to the wall each time. He never reaches
the wall, yet that is his limit postion. We could write

lim( Positionn ) = Wall

In our case lim( Sn ) = 2. Since this limit exists,
we say that the sum of the series
is 2, even though we can't really "do the sum."

Another example

Our first example was easy to understand because
there is a simple formula for the partial sums.
Now let's look at a more difficult example.

S = 1 + 1/4 + 1/9 + 1/16 + ... + 1/n2 + ....

We can compute some partial sums in an effort to
see what the limit might be:

This time it is not clear what is happening. The partial
sums are increasing, since we get one from another by
adding a positive number. But do they approach a limit?
Is there a number to which they get closer and closer
as we add more terms? If there is a limit what is it?
Can we compute it to some modest accuracy, say one or two decimal
places?

Problem

Use the "calculator" below to make an intelligent
guess about whether the limit S exists, and if
it does, what its value is, accurate to two decimal
places.

Series calculator
Number of terms:
Partial sum:

Notes

1. Computing the partial sums

We can read the definition of sum as follows. First,
create a variable s and set it to zero. This is a "container" in
which we will accumulate the partial sums. Then repeatedly add
1/i2 to s as i ranges from 1 to n.

The heart of the definition is the "repeatedly .... as i ranges
from 1 to n" part. Here the "..." is an action or sequence
of actions to be performed. This contruction is called a loop,
or, in more detail, a for loop.

2. Convergence

You probably concluded that the limit S exists and were able to
find a reasonably accurate value for it. However, there is
is a subtle problem because the partial sums do keep increasing
as n increases. True, they increase at a decreasing rate. But
is this enough to guarantee the existence of a limit, the "sum"
of the series? Now in fact the limit does exist, or, as we say,
the series converges. This, however, requires an argument
that we are not yet prepared to give.

It is important to understand clearly that there is a problem.
To this end consider the so-called harmonic series,

H = 1 + 1/2 + 1/3 + 1/4 + ....

Does it converge? See the next section for more
on this question. However, before going on, see what you can find by yourself.

3. Convergence criteria

Let's take a closer look at the question of whether the series

S = 1 + 1/4 + 1/9 + 1/16 + ... + 1/n2 + ....

converges. Since its terms are postive, the partial sums
form an increasing sequence:

S1 < S2 < S3 < ...

An increasing sequence behaves in one of two ways. Either
there is a number M which is bigger than all the terms of
the sequence, or else there is no such number. In the first
case we say that the sequence is bounded above.
The completeness axiom for the real numbers, the one that
guarantees that there are no holes or gaps in the number
line, also guarantees that such a sequence has a limit.

Theorem. A increasing sequence which is bounded
above by a number M converges to a limit L.
This limit is less than or equal to M.

In the second
case the sequence is unbounded: no matter what M we
choose, we can find an n such that Sn > M.
We say that it "tends to infinity," a fact we write as

lim( Sn ) = infinity

The main point of all this is that we can guarantee that a
series of positive terms converges if its partial sums are
bounded above. Can we do this with the sum S above?
If, for example, we can show that

Sn < 2

for all n, then we know (by virtue of the theorem) that the
sequence { Sn } converges and that its limit is less
than or equal to 2. The question is, therefore: how do we
show that such an inequality is true?

One of the key's to answerin this question
is to be found in the figure below. The
figure composed of yellow boxes is a model
for the partial sum of a series. Each box
has unit width. The height of the n-th box
is the magnitude of the n-th term. Thus the
area of the n-th box has the same magnitude
as the n-th term. Consequently the area of
the yellow figure is the magnitude of the
n-th partial sum. Now consider the green
figure, which we imagine extending infinitely
off to the right. ... to be continued.

4. Zeno's paradox

Our argument about taking smaller and smaller steps
toward the wall is really due to the Greek philosopher Zeno.
He imagined an arrow flying towards its target and
argued that since it would never reach it, that no motion
was possible. What is the flaw in his argument?